Linear Regression is a process by which the equation of a line is found that “best fits” a given set of data. The line of best fit approximates the best linear representation for your data. One very important aspect of a regression line is the relationship between the equation and the “science quantity” often represented by the slope of the line.

In order to solve problems involving linear regression, it is necessary to

Let's look at an example of linear regression by examining the data in the following table to discover the relationship between temperatures measured in Celsius (Centigrade) and Fahrenheit. [Remember that lines are named using the convention y vs. x whereas data tables are constructed as x | y.]

relate the linear equation of your model with its associated science formula to determine the "physical" meaning of the slope of this data's trend line

Step 1: First we will plot the data using a TI-83 graphing calculator. We will enter the data measured in degrees Fahrenheit in L1 and the temperatures measured in degrees Celsius in L2. Once the data is entered, your screen should look like the following:

After entering the data into the calculator, graph the data. The Fahrenheit data, listed in L1, represents the x-axis, and the Celsius data, listed in L2, represents the y-axis. Your screen should look like the following:

Step Two: Now we need to find a linear equation that models the data we have plotted. According to the calculator, our equation has the following properties:

Based on the graph and the equation information listed above, our correlation coefficient (r) is equal to 1. That means that our data perfectly models a linear function.

Step Four: Using the model from step two and the graph on our calculator from step three, we can trace along the graph and determine what temperature in degrees Celsius equals 98.6 °F, our body temperature.

This screen capture shows us that 98.6 °F (x-value) is equivalent to 37 °C (y-value).

Step Five: Consider our equation. The accepted formula used to convert Fahrenheit degrees to Celsius degrees is typically written as

Expressed in this form we can clearly see that our model's equation is indeed the same equation conventionally used to convert temperatures between these two measuring scales. Since our line's slope [] is a decimal, we know that the size of a "degree" on the two temperature scales is not the same; that is, these scales are not in a one-to-one correspondence -- 1 Celsius degree (Cº) does not equal 1 Fahrenheit degree (Fº).

Examining the slope in fraction form [] we can clearly see that the relationship between the two scales is such that from a given point on the line, you move up five degrees on the Celsius scale and right nine degrees on the Fahrenheit scale to arrive at the next point on the line. Or equivalently, when the temperature changes 9 Fº it only changes 5 Cº.

Obesity is a risk factor for the development of medical problems such as high blood pressure, high cholesterol, heart disease, and diabetes. How much a person can safely weigh depends on his or her height. Body Mass Index (BMI) is a method of examining a person's weight in accordance with his or her height.

According to Roche Pharmaceuticals, in Australia, a BMI of 30 or greater can create an increased risk of developing medical problems associated with obesity. The chart below shows height (in inches) and weight (in pounds) for individuals who have a BMI of 30.

A ball is rolled down a hallway and its position is recorded at five different times. Use the data given in the table shown below to predict the location of the ball at 12 seconds.

Time(seconds)

Position(meters)

1

9

2

12

4

17

6

21

8

26

What is your answer?A. 33 metersB. 34 metersC. 35 metersD. 36 meters

The data in the table below represents the apparent temperature (ºF) vs. the relative humidity (%) in a room whose actual temperature is 72 °F. Use this data to predict the apparent temperature when the relative humidity reaches 110%.

Relative Humidity(%)

Apparent Temperature(ºF)

0

64

10

65

20

67

30

68

40

70

50

71

60

72

70

73

80

74

90

75

100

76

What is your answer?A. Approximately 76.6°B. Approximately 77.7°C. Approximately 78.4°D. Approximately 65.7°

As you can see, this type of problem requires that you use a graphing calculator and a modeling approach. You must correctly enter the data into your calculator, graph the data, calculate an equation that best fits the data, graph that equation, and then make the prediction asked for in the problem. To better understand the data's behavior, you should endeavor to find the relationship between the equation of the line and the “physical” meaning if its slope. At the conclusion of the problem, you should always check for the reasonableness of your solutions.